LMFDB - Embedded newform 448.4.j.b.335.35

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,4,Mod(111,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.111"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 2])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.j (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [88] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$44$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			335.35
Character	$\chi$	$=$	448.335
Dual form			448.4.j.b.111.35

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.69197 - 4.69197i) q^{3} +(-9.81242 + 9.81242i) q^{5} +(17.7338 - 5.33982i) q^{7} -17.0292i q^{9} +(-30.5036 + 30.5036i) q^{11} +(-18.0708 - 18.0708i) q^{13} +92.0792i q^{15} +79.3893i q^{17} +(-75.4829 + 75.4829i) q^{19} +(58.1520 - 108.261i) q^{21} +105.136 q^{23} -67.5671i q^{25} +(46.7827 + 46.7827i) q^{27} +(19.3369 - 19.3369i) q^{29} -51.4882 q^{31} +286.244i q^{33} +(-121.615 + 226.408i) q^{35} +(12.1686 + 12.1686i) q^{37} -169.575 q^{39} -249.280 q^{41} +(-352.441 + 352.441i) q^{43} +(167.098 + 167.098i) q^{45} -56.2873 q^{47} +(285.973 - 189.390i) q^{49} +(372.492 + 372.492i) q^{51} +(362.373 + 362.373i) q^{53} -598.629i q^{55} +708.327i q^{57} +(302.126 + 302.126i) q^{59} +(-239.843 - 239.843i) q^{61} +(-90.9328 - 301.992i) q^{63} +354.636 q^{65} +(171.874 + 171.874i) q^{67} +(493.295 - 493.295i) q^{69} +1066.25 q^{71} -1120.92 q^{73} +(-317.023 - 317.023i) q^{75} +(-378.060 + 703.828i) q^{77} +770.664i q^{79} +898.795 q^{81} +(941.243 - 941.243i) q^{83} +(-779.000 - 779.000i) q^{85} -181.456i q^{87} +912.463 q^{89} +(-416.958 - 223.968i) q^{91} +(-241.581 + 241.581i) q^{93} -1481.34i q^{95} -1158.89i q^{97} +(519.452 + 519.452i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77}+ \cdots - 7592 q^{99}+O(q^{100}) $ 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	4.69197	−	4.69197i	0.902970	−	0.902970i	−0.0927217	−	0.995692i	$-0.529557\pi$
$3$	4.69197	−	4.69197i	0.902970	−	0.902970i	0.995692	+	0.0927217i	$0.0295567\pi$
$4$		0			0
$5$	−9.81242	+	9.81242i	−0.877649	+	0.877649i	−0.993291	−	0.115642i	$-0.963108\pi$
$5$	−9.81242	+	9.81242i	−0.877649	+	0.877649i	0.115642	+	0.993291i	$0.463108\pi$
$6$		0			0
$7$	17.7338	−	5.33982i	0.957533	−	0.288323i
$7$	17.7338	−	5.33982i	0.957533	−	0.288323i
$8$		0			0
$9$		−	17.0292i		−	0.630711i
$10$		0			0
$11$	−30.5036	+	30.5036i	−0.836108	+	0.836108i	−0.988344	−	0.152236i	$-0.951353\pi$
$11$	−30.5036	+	30.5036i	−0.836108	+	0.836108i	0.152236	+	0.988344i	$0.451353\pi$
$12$		0			0
$13$	−18.0708	−	18.0708i	−0.385534	−	0.385534i	0.487557	−	0.873091i	$-0.337888\pi$
$13$	−18.0708	−	18.0708i	−0.385534	−	0.385534i	−0.873091	+	0.487557i	$0.837888\pi$
$14$		0			0
$15$			92.0792i			1.58498i
$16$		0			0
$17$			79.3893i			1.13263i	0.824189	+	0.566315i	$0.191632\pi$
$17$			79.3893i			1.13263i	−0.824189	+	0.566315i	$0.808368\pi$
$18$		0			0
$19$	−75.4829	+	75.4829i	−0.911419	+	0.911419i	−0.996384	−	0.0849651i	$-0.972922\pi$
$19$	−75.4829	+	75.4829i	−0.911419	+	0.911419i	0.0849651	+	0.996384i	$0.472922\pi$
$20$		0			0
$21$	58.1520	−	108.261i	0.604277	−	1.12497i
$22$		0			0
$23$	105.136			0.953146			0.476573	−	0.879135i	$-0.341879\pi$
$23$	105.136			0.953146			0.476573	+	0.879135i	$0.341879\pi$
$24$		0			0
$25$		−	67.5671i		−	0.540536i
$26$		0			0
$27$	46.7827	+	46.7827i	0.333457	+	0.333457i
$28$		0			0
$29$	19.3369	−	19.3369i	0.123820	−	0.123820i	−0.642481	−	0.766301i	$-0.722095\pi$
$29$	19.3369	−	19.3369i	0.123820	−	0.123820i	0.766301	+	0.642481i	$0.222095\pi$
$30$		0			0
$31$	−51.4882			−0.298308			−0.149154	−	0.988814i	$-0.547655\pi$
$31$	−51.4882			−0.298308			−0.149154	+	0.988814i	$0.547655\pi$
$32$		0			0
$33$			286.244i			1.50996i
$34$		0			0
$35$	−121.615	+	226.408i	−0.587332	+	1.09342i
$36$		0			0
$37$	12.1686	+	12.1686i	0.0540678	+	0.0540678i	0.733624	−	0.679556i	$-0.237828\pi$
$37$	12.1686	+	12.1686i	0.0540678	+	0.0540678i	−0.679556	+	0.733624i	$0.737828\pi$
$38$		0			0
$39$	−169.575			−0.696251
$40$		0			0
$41$	−249.280			−0.949535			−0.474768	−	0.880111i	$-0.657468\pi$
$41$	−249.280			−0.949535			−0.474768	+	0.880111i	$0.657468\pi$
$42$		0			0
$43$	−352.441	+	352.441i	−1.24992	+	1.24992i	−0.294171	+	0.955753i	$0.595044\pi$
$43$	−352.441	+	352.441i	−1.24992	+	1.24992i	−0.955753	+	0.294171i	$0.904956\pi$
$44$		0			0
$45$	167.098	+	167.098i	0.553543	+	0.553543i
$46$		0			0
$47$	−56.2873			−0.174688			−0.0873440	−	0.996178i	$-0.527838\pi$
$47$	−56.2873			−0.174688			−0.0873440	+	0.996178i	$0.527838\pi$
$48$		0			0
$49$	285.973	−	189.390i	0.833740	−	0.552158i
$50$		0			0
$51$	372.492	+	372.492i	1.02273	+	1.02273i
$52$		0			0
$53$	362.373	+	362.373i	0.939166	+	0.939166i	0.998253	−	0.0590864i	$-0.0188188\pi$
$53$	362.373	+	362.373i	0.939166	+	0.939166i	−0.0590864	+	0.998253i	$0.518819\pi$
$54$		0			0
$55$		−	598.629i		−	1.46762i
$56$		0			0
$57$			708.327i			1.64597i
$58$		0			0
$59$	302.126	+	302.126i	0.666670	+	0.666670i	0.956944	−	0.290274i	$-0.0937464\pi$
$59$	302.126	+	302.126i	0.666670	+	0.666670i	−0.290274	+	0.956944i	$0.593746\pi$
$60$		0			0
$61$	−239.843	−	239.843i	−0.503421	−	0.503421i	0.409078	−	0.912499i	$-0.365850\pi$
$61$	−239.843	−	239.843i	−0.503421	−	0.503421i	−0.912499	+	0.409078i	$0.865850\pi$
$62$		0			0
$63$	−90.9328	−	301.992i	−0.181848	−	0.603927i
$64$		0			0
$65$	354.636			0.676727
$66$		0			0
$67$	171.874	+	171.874i	0.313399	+	0.313399i	0.846225	−	0.532826i	$-0.178870\pi$
$67$	171.874	+	171.874i	0.313399	+	0.313399i	−0.532826	+	0.846225i	$0.678870\pi$
$68$		0			0
$69$	493.295	−	493.295i	0.860663	−	0.860663i
$70$		0			0
$71$	1066.25			1.78225			0.891127	−	0.453754i	$-0.149915\pi$
$71$	1066.25			1.78225			0.891127	+	0.453754i	$0.149915\pi$
$72$		0			0
$73$	−1120.92			−1.79717			−0.898587	−	0.438794i	$-0.855406\pi$
$73$	−1120.92			−1.79717			−0.898587	+	0.438794i	$0.855406\pi$
$74$		0			0
$75$	−317.023	−	317.023i	−0.488088	−	0.488088i
$76$		0			0
$77$	−378.060	+	703.828i	−0.559532	+	1.04167i
$78$		0			0
$79$			770.664i			1.09755i	0.835970	+	0.548775i	$0.184906\pi$
$79$			770.664i			1.09755i	−0.835970	+	0.548775i	$0.815094\pi$
$80$		0			0
$81$	898.795			1.23291
$82$		0			0
$83$	941.243	−	941.243i	1.24476	−	1.24476i	0.286753	−	0.958004i	$-0.407424\pi$
$83$	941.243	−	941.243i	1.24476	−	1.24476i	0.958004	−	0.286753i	$-0.0925760\pi$
$84$		0			0
$85$	−779.000	−	779.000i	−0.994052	−	0.994052i
$86$		0			0
$87$		−	181.456i		−	0.223611i
$88$		0			0
$89$	912.463			1.08675			0.543376	−	0.839490i	$-0.317146\pi$
$89$	912.463			1.08675			0.543376	+	0.839490i	$0.317146\pi$
$90$		0			0
$91$	−416.958	−	223.968i	−0.480319	−	0.258003i
$92$		0			0
$93$	−241.581	+	241.581i	−0.269363	+	0.269363i
$94$		0			0
$95$		−	1481.34i		−	1.59981i
$96$		0			0
$97$		−	1158.89i		−	1.21306i	−0.795060	−	0.606531i	$-0.792561\pi$
$97$		−	1158.89i		−	1.21306i	0.795060	−	0.606531i	$-0.207439\pi$
$98$		0			0
$99$	519.452	+	519.452i	0.527343	+	0.527343i
$100$		0			0
$101$	−464.657	+	464.657i	−0.457773	+	0.457773i	−0.897924	−	0.440151i	$-0.854925\pi$
$101$	−464.657	+	464.657i	−0.457773	+	0.457773i	0.440151	+	0.897924i	$0.354925\pi$
$102$		0			0
$103$			1831.71i			1.75227i	0.482070	+	0.876133i	$0.339885\pi$
$103$			1831.71i			1.75227i	−0.482070	+	0.876133i	$0.660115\pi$
$104$		0			0
$105$	491.686	+	1632.91i	0.456987	+	1.51767i
$106$		0			0
$107$	−299.986	+	299.986i	−0.271035	+	0.271035i	−0.829517	−	0.558482i	$-0.811384\pi$
$107$	−299.986	+	299.986i	−0.271035	+	0.271035i	0.558482	+	0.829517i	$0.311384\pi$
$108$		0			0
$109$	−403.065	+	403.065i	−0.354189	+	0.354189i	−0.861666	−	0.507476i	$-0.830578\pi$
$109$	−403.065	+	403.065i	−0.354189	+	0.354189i	0.507476	+	0.861666i	$0.330578\pi$
$110$		0			0
$111$	114.190			0.0976432
$112$		0			0
$113$	422.142			0.351431			0.175716	−	0.984441i	$-0.443776\pi$
$113$	422.142			0.351431			0.175716	+	0.984441i	$0.443776\pi$
$114$		0			0
$115$	−1031.64	+	1031.64i	−0.836528	+	0.836528i
$116$		0			0
$117$	−307.731	+	307.731i	−0.243160	+	0.243160i
$118$		0			0
$119$	423.924	+	1407.87i	0.326564	+	1.08453i
$120$		0			0
$121$		−	529.943i		−	0.398154i
$122$		0			0
$123$	−1169.61	+	1169.61i	−0.857402	+	0.857402i
$124$		0			0
$125$	−563.556	−	563.556i	−0.403248	−	0.403248i
$126$		0			0
$127$		−	2251.58i		−	1.57319i	−0.617470	−	0.786595i	$-0.711842\pi$
$127$		−	2251.58i		−	1.57319i	0.617470	−	0.786595i	$-0.288158\pi$
$128$		0			0
$129$			3307.29i			2.25729i
$130$		0			0
$131$	−1604.63	+	1604.63i	−1.07021	+	1.07021i	−0.0728652	+	0.997342i	$0.523214\pi$
$131$	−1604.63	+	1604.63i	−1.07021	+	1.07021i	−0.997342	+	0.0728652i	$0.976786\pi$
$132$		0			0
$133$	−935.531	+	1741.66i	−0.609931	+	1.13550i
$134$		0			0
$135$	−918.103			−0.585317
$136$		0			0
$137$			429.819i			0.268043i	0.990978	+	0.134022i	$0.0427892\pi$
$137$			429.819i			0.268043i	−0.990978	+	0.134022i	$0.957211\pi$
$138$		0			0
$139$	−1480.20	−	1480.20i	−0.903230	−	0.903230i	0.0924842	−	0.995714i	$-0.470519\pi$
$139$	−1480.20	−	1480.20i	−0.903230	−	0.903230i	−0.995714	+	0.0924842i	$0.970519\pi$
$140$		0			0
$141$	−264.098	+	264.098i	−0.157738	+	0.157738i
$142$		0			0
$143$	1102.45			0.644696
$144$		0			0
$145$			379.483i			0.217341i
$146$		0			0
$147$	453.163	−	2230.39i	0.254260	−	1.25142i
$148$		0			0
$149$	−1546.67	−	1546.67i	−0.850391	−	0.850391i	0.139790	−	0.990181i	$-0.455357\pi$
$149$	−1546.67	−	1546.67i	−0.850391	−	0.850391i	−0.990181	+	0.139790i	$0.955357\pi$
$150$		0			0
$151$	465.738			0.251001			0.125501	−	0.992094i	$-0.459946\pi$
$151$	465.738			0.251001			0.125501	+	0.992094i	$0.459946\pi$
$152$		0			0
$153$	1351.94			0.714363
$154$		0			0
$155$	505.223	−	505.223i	0.261810	−	0.261810i
$156$		0			0
$157$	−413.093	−	413.093i	−0.209990	−	0.209990i	0.594273	−	0.804263i	$-0.297440\pi$
$157$	−413.093	−	413.093i	−0.209990	−	0.209990i	−0.804263	+	0.594273i	$0.797440\pi$
$158$		0			0
$159$	3400.49			1.69608
$160$		0			0
$161$	1864.46	−	561.407i	0.912669	−	0.274814i
$162$		0			0
$163$	1386.17	+	1386.17i	0.666092	+	0.666092i	0.956809	−	0.290717i	$-0.0938937\pi$
$163$	1386.17	+	1386.17i	0.666092	+	0.666092i	−0.290717	+	0.956809i	$0.593894\pi$
$164$		0			0
$165$	−2808.75	−	2808.75i	−1.32522	−	1.32522i
$166$		0			0
$167$		−	1007.83i		−	0.466995i	−0.972357	−	0.233498i	$-0.924983\pi$
$167$		−	1007.83i		−	0.466995i	0.972357	−	0.233498i	$-0.0750171\pi$
$168$		0			0
$169$		−	1543.89i		−	0.702728i
$170$		0			0
$171$	1285.41	+	1285.41i	0.574842	+	0.574842i
$172$		0			0
$173$	−1409.91	−	1409.91i	−0.619616	−	0.619616i	0.325817	−	0.945433i	$-0.394361\pi$
$173$	−1409.91	−	1409.91i	−0.619616	−	0.619616i	−0.945433	+	0.325817i	$0.894361\pi$
$174$		0			0
$175$	−360.796	−	1198.22i	−0.155849	−	0.517582i
$176$		0			0
$177$	2835.14			1.20397
$178$		0			0
$179$	−1896.74	−	1896.74i	−0.792006	−	0.792006i	0.189814	−	0.981820i	$-0.439211\pi$
$179$	−1896.74	−	1896.74i	−0.792006	−	0.792006i	−0.981820	+	0.189814i	$0.939211\pi$
$180$		0			0
$181$	1960.65	−	1960.65i	0.805161	−	0.805161i	−0.178736	−	0.983897i	$-0.557201\pi$
$181$	1960.65	−	1960.65i	0.805161	−	0.805161i	0.983897	+	0.178736i	$0.0572009\pi$
$182$		0			0
$183$	−2250.67			−0.909149
$184$		0			0
$185$	−238.807			−0.0949051
$186$		0			0
$187$	−2421.66	−	2421.66i	−0.947002	−	0.947002i
$188$		0			0
$189$	1079.45	+	579.823i	0.415440	+	0.223153i
$190$		0			0
$191$			727.067i			0.275438i	0.990471	+	0.137719i	$0.0439771\pi$
$191$			727.067i			0.275438i	−0.990471	+	0.137719i	$0.956023\pi$
$192$		0			0
$193$	−2769.63			−1.03296			−0.516482	−	0.856298i	$-0.672759\pi$
$193$	−2769.63			−1.03296			−0.516482	+	0.856298i	$0.672759\pi$
$194$		0			0
$195$	1663.94	−	1663.94i	0.611064	−	0.611064i
$196$		0			0
$197$	−1735.01	−	1735.01i	−0.627484	−	0.627484i	0.319951	−	0.947434i	$-0.396334\pi$
$197$	−1735.01	−	1735.01i	−0.627484	−	0.627484i	−0.947434	+	0.319951i	$0.896334\pi$
$198$		0			0
$199$		−	913.138i		−	0.325280i	−0.986686	−	0.162640i	$-0.947999\pi$
$199$		−	913.138i		−	0.325280i	0.986686	−	0.162640i	$-0.0520009\pi$
$200$		0			0
$201$	1612.85			0.565979
$202$		0			0
$203$	239.660	−	446.171i	0.0828614	−	0.154262i
$204$		0			0
$205$	2446.04	−	2446.04i	0.833359	−	0.833359i
$206$		0			0
$207$		−	1790.38i		−	0.601160i
$208$		0			0
$209$		−	4605.00i		−	1.52409i
$210$		0			0
$211$	993.073	+	993.073i	0.324009	+	0.324009i	0.850303	−	0.526294i	$-0.176419\pi$
$211$	993.073	+	993.073i	0.324009	+	0.324009i	−0.526294	+	0.850303i	$0.676419\pi$
$212$		0			0
$213$	5002.79	−	5002.79i	1.60932	−	1.60932i
$214$		0			0
$215$		−	6916.59i		−	2.19399i
$216$		0			0
$217$	−913.079	+	274.937i	−0.285640	+	0.0860090i
$218$		0			0
$219$	−5259.33	+	5259.33i	−1.62280	+	1.62280i
$220$		0			0
$221$	1434.63	−	1434.63i	0.436667	−	0.436667i
$222$		0			0
$223$	−2729.54			−0.819656			−0.409828	−	0.912163i	$-0.634411\pi$
$223$	−2729.54			−0.819656			−0.409828	+	0.912163i	$0.634411\pi$
$224$		0			0
$225$	−1150.61			−0.340922
$226$		0			0
$227$	−2957.54	+	2957.54i	−0.864752	+	0.864752i	−0.991886	−	0.127134i	$-0.959422\pi$
$227$	−2957.54	+	2957.54i	−0.864752	+	0.864752i	0.127134	+	0.991886i	$0.459422\pi$
$228$		0			0
$229$	676.038	−	676.038i	0.195082	−	0.195082i	−0.602806	−	0.797888i	$-0.705951\pi$
$229$	676.038	−	676.038i	0.195082	−	0.195082i	0.797888	+	0.602806i	$0.205951\pi$
$230$		0			0
$231$	1528.49	+	5076.19i	0.435357	+	1.44584i
$232$		0			0
$233$		−	1728.30i		−	0.485943i	−0.970033	−	0.242972i	$-0.921878\pi$
$233$		−	1728.30i		−	0.485943i	0.970033	−	0.242972i	$-0.0781222\pi$
$234$		0			0
$235$	552.314	−	552.314i	0.153315	−	0.153315i
$236$		0			0
$237$	3615.94	+	3615.94i	0.991056	+	0.991056i
$238$		0			0
$239$		−	2821.13i		−	0.763530i	−0.924259	−	0.381765i	$-0.875316\pi$
$239$		−	2821.13i		−	0.763530i	0.924259	−	0.381765i	$-0.124684\pi$
$240$		0			0
$241$		−	413.034i		−	0.110398i	−0.998475	−	0.0551989i	$-0.982421\pi$
$241$		−	413.034i		−	0.110398i	0.998475	−	0.0551989i	$-0.0175793\pi$
$242$		0			0
$243$	2953.99	−	2953.99i	0.779828	−	0.779828i
$244$		0			0
$245$	−947.709	+	4664.46i	−0.247130	+	1.21633i
$246$		0			0
$247$	2728.07			0.702765
$248$		0			0
$249$		−	8832.58i		−	2.24796i
$250$		0			0
$251$	5128.40	+	5128.40i	1.28965	+	1.28965i	0.934999	+	0.354650i	$0.115400\pi$
$251$	5128.40	+	5128.40i	1.28965	+	1.28965i	0.354650	+	0.934999i	$0.384600\pi$
$252$		0			0
$253$	−3207.03	+	3207.03i	−0.796933	+	0.796933i
$254$		0			0
$255$	−7310.10			−1.79520
$256$		0			0
$257$			3638.50i			0.883125i	0.897231	+	0.441562i	$0.145576\pi$
$257$			3638.50i			0.883125i	−0.897231	+	0.441562i	$0.854424\pi$
$258$		0			0
$259$	280.773	+	150.817i	0.0673607	+	0.0361827i
$260$		0			0
$261$	−329.292	−	329.292i	−0.0780945	−	0.0780945i
$262$		0			0
$263$	345.728			0.0810588			0.0405294	−	0.999178i	$-0.487096\pi$
$263$	345.728			0.0810588			0.0405294	+	0.999178i	$0.487096\pi$
$264$		0			0
$265$	−7111.52			−1.64852
$266$		0			0
$267$	4281.25	−	4281.25i	0.981305	−	0.981305i
$268$		0			0
$269$	3684.78	+	3684.78i	0.835185	+	0.835185i	0.988221	−	0.153036i	$-0.0489049\pi$
$269$	3684.78	+	3684.78i	0.835185	+	0.835185i	−0.153036	+	0.988221i	$0.548905\pi$
$270$		0			0
$271$	7814.42			1.75163			0.875816	−	0.482646i	$-0.160324\pi$
$271$	7814.42			1.75163			0.875816	+	0.482646i	$0.160324\pi$
$272$		0			0
$273$	−3007.21	+	905.501i	−0.666683	+	0.200745i
$274$		0			0
$275$	2061.04	+	2061.04i	0.451947	+	0.451947i
$276$		0			0
$277$	−2761.15	−	2761.15i	−0.598921	−	0.598921i	0.341104	−	0.940026i	$-0.389199\pi$
$277$	−2761.15	−	2761.15i	−0.598921	−	0.598921i	−0.940026	+	0.341104i	$0.889199\pi$
$278$		0			0
$279$			876.802i			0.188146i
$280$		0			0
$281$			6031.16i			1.28039i	0.768214	+	0.640193i	$0.221146\pi$
$281$			6031.16i			1.28039i	−0.768214	+	0.640193i	$0.778854\pi$
$282$		0			0
$283$	2874.34	+	2874.34i	0.603753	+	0.603753i	0.941306	−	0.337554i	$-0.109599\pi$
$283$	2874.34	+	2874.34i	0.603753	+	0.603753i	−0.337554	+	0.941306i	$0.609599\pi$
$284$		0			0
$285$	−6950.40	−	6950.40i	−1.44458	−	1.44458i
$286$		0			0
$287$	−4420.67	+	1331.11i	−0.909212	+	0.273773i
$288$		0			0
$289$	−1389.65			−0.282852
$290$		0			0
$291$	−5437.46	−	5437.46i	−1.09536	−	1.09536i
$292$		0			0
$293$	−3169.82	+	3169.82i	−0.632023	+	0.632023i	−0.948575	−	0.316552i	$-0.897475\pi$
$293$	−3169.82	+	3169.82i	−0.632023	+	0.632023i	0.316552	+	0.948575i	$0.397475\pi$
$294$		0			0
$295$	−5929.18			−1.17020
$296$		0			0
$297$	−2854.09			−0.557612
$298$		0			0
$299$	−1899.89	−	1899.89i	−0.367470	−	0.367470i
$300$		0			0
$301$	−4368.13	+	8132.07i	−0.836462	+	1.55723i
$302$		0			0
$303$			4360.31i			0.826711i
$304$		0			0
$305$	4706.87			0.883655
$306$		0			0
$307$	804.857	−	804.857i	0.149627	−	0.149627i	−0.628324	−	0.777952i	$-0.716259\pi$
$307$	804.857	−	804.857i	0.149627	−	0.149627i	0.777952	+	0.628324i	$0.216259\pi$
$308$		0			0
$309$	8594.31	+	8594.31i	1.58224	+	1.58224i
$310$		0			0
$311$		−	3145.24i		−	0.573473i	−0.958009	−	0.286736i	$-0.907430\pi$
$311$		−	3145.24i		−	0.573473i	0.958009	−	0.286736i	$-0.0925704\pi$
$312$		0			0
$313$	4781.18			0.863414			0.431707	−	0.902014i	$-0.357911\pi$
$313$	4781.18			0.863414			0.431707	+	0.902014i	$0.357911\pi$
$314$		0			0
$315$	3855.54	+	2071.00i	0.689635	+	0.370437i
$316$		0			0
$317$	662.345	−	662.345i	0.117353	−	0.117353i	−0.645991	−	0.763345i	$-0.723556\pi$
$317$	662.345	−	662.345i	0.117353	−	0.117353i	0.763345	+	0.645991i	$0.223556\pi$
$318$		0			0
$319$			1179.69i			0.207053i
$320$		0			0
$321$			2815.05i			0.489474i
$322$		0			0
$323$	−5992.53	−	5992.53i	−1.03230	−	1.03230i
$324$		0			0
$325$	−1220.99	+	1220.99i	−0.208395	+	0.208395i
$326$		0			0
$327$			3782.34i			0.639645i
$328$		0			0
$329$	−998.185	+	300.564i	−0.167270	+	0.0503666i
$330$		0			0
$331$	1946.43	−	1946.43i	0.323219	−	0.323219i	−0.526781	−	0.850001i	$-0.676601\pi$
$331$	1946.43	−	1946.43i	0.323219	−	0.323219i	0.850001	+	0.526781i	$0.176601\pi$
$332$		0			0
$333$	207.222	−	207.222i	0.0341011	−	0.0341011i
$334$		0			0
$335$	−3372.99			−0.550108
$336$		0			0
$337$	−6957.87			−1.12469			−0.562343	−	0.826904i	$-0.690100\pi$
$337$	−6957.87			−1.12469			−0.562343	+	0.826904i	$0.690100\pi$
$338$		0			0
$339$	1980.68	−	1980.68i	0.317332	−	0.317332i
$340$		0			0
$341$	1570.58	−	1570.58i	0.249418	−	0.249418i
$342$		0			0
$343$	4060.06	−	4885.64i	0.639134	−	0.769096i
$344$		0			0
$345$			9680.83i			1.51072i
$346$		0			0
$347$	4059.62	−	4059.62i	0.628046	−	0.628046i	−0.319530	−	0.947576i	$-0.603525\pi$
$347$	4059.62	−	4059.62i	0.628046	−	0.628046i	0.947576	+	0.319530i	$0.103525\pi$
$348$		0			0
$349$	5227.72	+	5227.72i	0.801814	+	0.801814i	0.983379	−	0.181565i	$-0.0581162\pi$
$349$	5227.72	+	5227.72i	0.801814	+	0.801814i	−0.181565	+	0.983379i	$0.558116\pi$
$350$		0			0
$351$		−	1690.80i		−	0.257118i
$352$		0			0
$353$		−	2069.94i		−	0.312101i	−0.987749	−	0.156050i	$-0.950124\pi$
$353$		−	2069.94i		−	0.312101i	0.987749	−	0.156050i	$-0.0498762\pi$
$354$		0			0
$355$	−10462.4	+	10462.4i	−1.56419	+	1.56419i
$356$		0			0
$357$	8594.73	+	4616.65i	1.27418	+	0.684423i
$358$		0			0
$359$	1926.22			0.283180			0.141590	−	0.989925i	$-0.454778\pi$
$359$	1926.22			0.283180			0.141590	+	0.989925i	$0.454778\pi$
$360$		0			0
$361$		−	4536.33i		−	0.661369i
$362$		0			0
$363$	−2486.48	−	2486.48i	−0.359521	−	0.359521i
$364$		0			0
$365$	10998.9	−	10998.9i	1.57729	−	1.57729i
$366$		0			0
$367$	8089.10			1.15054			0.575269	−	0.817964i	$-0.304897\pi$
$367$	8089.10			1.15054			0.575269	+	0.817964i	$0.304897\pi$
$368$		0			0
$369$			4245.03i			0.598882i
$370$		0			0
$371$	8361.25	+	4491.24i	1.17007	+	0.628500i
$372$		0			0
$373$	9078.78	+	9078.78i	1.26027	+	1.26027i	0.950961	+	0.309311i	$0.100098\pi$
$373$	9078.78	+	9078.78i	1.26027	+	1.26027i	0.309311	+	0.950961i	$0.399902\pi$
$374$		0			0
$375$	−5288.38			−0.728242
$376$		0			0
$377$	−698.866			−0.0954733
$378$		0			0
$379$	2825.46	−	2825.46i	0.382939	−	0.382939i	−0.489221	−	0.872160i	$-0.662719\pi$
$379$	2825.46	−	2825.46i	0.382939	−	0.382939i	0.872160	+	0.489221i	$0.162719\pi$
$380$		0			0
$381$	−10564.3	−	10564.3i	−1.42054	−	1.42054i
$382$		0			0
$383$	−826.949			−0.110327			−0.0551634	−	0.998477i	$-0.517568\pi$
$383$	−826.949			−0.110327			−0.0551634	+	0.998477i	$0.517568\pi$
$384$		0			0
$385$	−3196.57	−	10615.9i	−0.423148	−	1.40529i
$386$		0			0
$387$	6001.78	+	6001.78i	0.788341	+	0.788341i
$388$		0			0
$389$	1617.43	+	1617.43i	0.210815	+	0.210815i	0.804614	−	0.593799i	$-0.202373\pi$
$389$	1617.43	+	1617.43i	0.210815	+	0.210815i	−0.593799	+	0.804614i	$0.702373\pi$
$390$		0			0
$391$			8346.67i			1.07956i
$392$		0			0
$393$			15057.8i			1.93273i
$394$		0			0
$395$	−7562.08	−	7562.08i	−0.963265	−	0.963265i
$396$		0			0
$397$	−3675.80	−	3675.80i	−0.464693	−	0.464693i	0.435497	−	0.900190i	$-0.356573\pi$
$397$	−3675.80	−	3675.80i	−0.464693	−	0.464693i	−0.900190	+	0.435497i	$0.856573\pi$
$398$		0			0
$399$	3782.34	+	12561.3i	0.474571	+	1.57607i
$400$		0			0
$401$	−910.404			−0.113375			−0.0566875	−	0.998392i	$-0.518054\pi$
$401$	−910.404			−0.113375			−0.0566875	+	0.998392i	$0.518054\pi$
$402$		0			0
$403$	930.432	+	930.432i	0.115008	+	0.115008i
$404$		0			0
$405$	−8819.35	+	8819.35i	−1.08207	+	1.08207i
$406$		0			0
$407$	−742.374			−0.0904130
$408$		0			0
$409$	11141.8			1.34701			0.673507	−	0.739181i	$-0.264787\pi$
$409$	11141.8			1.34701			0.673507	+	0.739181i	$0.264787\pi$
$410$		0			0
$411$	2016.70	+	2016.70i	0.242035	+	0.242035i
$412$		0			0
$413$	6971.14	+	3744.54i	0.830575	+	0.446142i
$414$		0			0
$415$			18471.7i			2.18492i
$416$		0			0
$417$	−13890.1			−1.63118
$418$		0			0
$419$	−1124.11	+	1124.11i	−0.131065	+	0.131065i	−0.769596	−	0.638531i	$-0.779542\pi$
$419$	−1124.11	+	1124.11i	−0.131065	+	0.131065i	0.638531	+	0.769596i	$0.279542\pi$
$420$		0			0
$421$	−10443.7	−	10443.7i	−1.20902	−	1.20902i	−0.971345	−	0.237674i	$-0.923615\pi$
$421$	−10443.7	−	10443.7i	−1.20902	−	1.20902i	−0.237674	−	0.971345i	$-0.576385\pi$
$422$		0			0
$423$			958.527i			0.110178i
$424$		0			0
$425$	5364.10			0.612228
$426$		0			0
$427$	−5534.03	−	2972.60i	−0.627191	−	0.336895i
$428$		0			0
$429$	5172.66	−	5172.66i	0.582141	−	0.582141i
$430$		0			0
$431$			9708.79i			1.08505i	0.840040	+	0.542524i	$0.182531\pi$
$431$			9708.79i			1.08505i	−0.840040	+	0.542524i	$0.817469\pi$
$432$		0			0
$433$			473.727i			0.0525771i	0.999654	+	0.0262885i	$0.00836886\pi$
$433$			473.727i			0.0525771i	−0.999654	+	0.0262885i	$0.991631\pi$
$434$		0			0
$435$	1780.53	+	1780.53i	0.196252	+	0.196252i
$436$		0			0
$437$	−7935.96	+	7935.96i	−0.868716	+	0.868716i
$438$		0			0
$439$		−	4168.28i		−	0.453169i	−0.973992	−	0.226584i	$-0.927244\pi$
$439$		−	4168.28i		−	0.453169i	0.973992	−	0.226584i	$-0.0727559\pi$
$440$		0			0
$441$	−3225.16	−	4869.89i	−0.348252	−	0.525849i
$442$		0			0
$443$	7613.87	−	7613.87i	0.816583	−	0.816583i	−0.169029	−	0.985611i	$-0.554063\pi$
$443$	7613.87	−	7613.87i	0.816583	−	0.816583i	0.985611	+	0.169029i	$0.0540630\pi$
$444$		0			0
$445$	−8953.47	+	8953.47i	−0.953787	+	0.953787i
$446$		0			0
$447$	−14513.9			−1.53576
$448$		0			0
$449$	7793.71			0.819172			0.409586	−	0.912272i	$-0.365673\pi$
$449$	7793.71			0.819172			0.409586	+	0.912272i	$0.365673\pi$
$450$		0			0
$451$	7603.93	−	7603.93i	0.793914	−	0.793914i
$452$		0			0
$453$	2185.23	−	2185.23i	0.226647	−	0.226647i
$454$		0			0
$455$	6289.04	−	1893.69i	0.647988	−	0.195116i
$456$		0			0
$457$		−	3142.19i		−	0.321632i	−0.986984	−	0.160816i	$-0.948587\pi$
$457$		−	3142.19i		−	0.321632i	0.986984	−	0.160816i	$-0.0514125\pi$
$458$		0			0
$459$	−3714.05	+	3714.05i	−0.377684	+	0.377684i
$460$		0			0
$461$	−3018.31	−	3018.31i	−0.304939	−	0.304939i	0.538004	−	0.842943i	$-0.319179\pi$
$461$	−3018.31	−	3018.31i	−0.304939	−	0.304939i	−0.842943	+	0.538004i	$0.819179\pi$
$462$		0			0
$463$			2582.65i			0.259236i	0.991564	+	0.129618i	$0.0413750\pi$
$463$			2582.65i			0.259236i	−0.991564	+	0.129618i	$0.958625\pi$
$464$		0			0
$465$		−	4740.99i		−	0.472813i
$466$		0			0
$467$	−1290.58	+	1290.58i	−0.127882	+	0.127882i	−0.768151	−	0.640269i	$-0.778823\pi$
$467$	−1290.58	+	1290.58i	−0.127882	+	0.127882i	0.640269	+	0.768151i	$0.278823\pi$
$468$		0			0
$469$	3965.74	+	2130.19i	0.390450	+	0.209729i
$470$		0			0
$471$	−3876.44			−0.379229
$472$		0			0
$473$		−	21501.5i		−	2.09014i
$474$		0			0
$475$	5100.16	+	5100.16i	0.492655	+	0.492655i
$476$		0			0
$477$	6170.93	−	6170.93i	0.592343	−	0.592343i
$478$		0			0
$479$	19407.1			1.85121			0.925607	−	0.378485i	$-0.123555\pi$
$479$	19407.1			1.85121			0.925607	+	0.378485i	$0.123555\pi$
$480$		0			0
$481$		−	439.793i		−	0.0416899i
$482$		0			0
$483$	6113.87	−	11382.1i	0.575964	−	1.07226i
$484$		0			0
$485$	11371.5	+	11371.5i	1.06464	+	1.06464i
$486$		0			0
$487$	−176.677			−0.0164395			−0.00821973	−	0.999966i	$-0.502616\pi$
$487$	−176.677			−0.0164395			−0.00821973	+	0.999966i	$0.502616\pi$
$488$		0			0
$489$	13007.7			1.20292
$490$		0			0
$491$	2146.08	−	2146.08i	0.197253	−	0.197253i	−0.601568	−	0.798821i	$-0.705457\pi$
$491$	2146.08	−	2146.08i	0.197253	−	0.197253i	0.798821	+	0.601568i	$0.205457\pi$
$492$		0			0
$493$	1535.14	+	1535.14i	0.140242	+	0.140242i
$494$		0			0
$495$	−10194.2			−0.925644
$496$		0			0
$497$	18908.5	−	5693.56i	1.70657	−	0.513865i
$498$		0			0
$499$	2551.77	+	2551.77i	0.228924	+	0.228924i	0.812243	−	0.583319i	$-0.198246\pi$
$499$	2551.77	+	2551.77i	0.228924	+	0.228924i	−0.583319	+	0.812243i	$0.698246\pi$
$500$		0			0
$501$	−4728.71	−	4728.71i	−0.421683	−	0.421683i
$502$		0			0
$503$			17403.8i			1.54273i	0.636390	+	0.771367i	$0.280427\pi$
$503$			17403.8i			1.54273i	−0.636390	+	0.771367i	$0.719573\pi$
$504$		0			0
$505$		−	9118.81i		−	0.803528i
$506$		0			0
$507$	−7243.90	−	7243.90i	−0.634542	−	0.634542i
$508$		0			0
$509$	13392.4	+	13392.4i	1.16622	+	1.16622i	0.983088	+	0.183136i	$0.0586248\pi$
$509$	13392.4	+	13392.4i	1.16622	+	1.16622i	0.183136	+	0.983088i	$0.441375\pi$
$510$		0			0
$511$	−19878.1	+	5985.51i	−1.72085	+	0.518167i
$512$		0			0
$513$	−7062.59			−0.607838
$514$		0			0
$515$	−17973.5	−	17973.5i	−1.53787	−	1.53787i
$516$		0			0
$517$	1716.97	−	1716.97i	0.146058	−	0.146058i
$518$		0			0
$519$	−13230.5			−1.11899
$520$		0			0
$521$	−4917.34			−0.413498			−0.206749	−	0.978394i	$-0.566288\pi$
$521$	−4917.34			−0.413498			−0.206749	+	0.978394i	$0.566288\pi$
$522$		0			0
$523$	2883.18	+	2883.18i	0.241057	+	0.241057i	0.817287	−	0.576230i	$-0.195477\pi$
$523$	2883.18	+	2883.18i	0.241057	+	0.241057i	−0.576230	+	0.817287i	$0.695477\pi$
$524$		0			0
$525$	−7314.85	−	3929.16i	−0.608088	−	0.326634i
$526$		0			0
$527$		−	4087.61i		−	0.337873i
$528$		0			0
$529$	−1113.43			−0.0915120
$530$		0			0
$531$	5144.97	−	5144.97i	0.420476	−	0.420476i
$532$		0			0
$533$	4504.68	+	4504.68i	0.366078	+	0.366078i
$534$		0			0
$535$		−	5887.18i		−	0.475748i
$536$		0			0
$537$	−17798.9			−1.43032
$538$		0			0
$539$	−2946.12	+	14500.3i	−0.235433	+	1.15876i
$540$		0			0
$541$	−13975.1	+	13975.1i	−1.11060	+	1.11060i	−0.117536	+	0.993069i	$0.537500\pi$
$541$	−13975.1	+	13975.1i	−1.11060	+	1.11060i	−0.993069	+	0.117536i	$0.962500\pi$
$542$		0			0
$543$		−	18398.6i		−	1.45407i
$544$		0			0
$545$		−	7910.08i		−	0.621708i
$546$		0			0
$547$	16130.8	+	16130.8i	1.26088	+	1.26088i	0.950666	+	0.310217i	$0.100402\pi$
$547$	16130.8	+	16130.8i	1.26088	+	1.26088i	0.310217	+	0.950666i	$0.399598\pi$
$548$		0			0
$549$	−4084.33	+	4084.33i	−0.317513	+	0.317513i
$550$		0			0
$551$			2919.21i			0.225703i
$552$		0			0
$553$	4115.21	+	13666.8i	0.316449	+	1.05094i
$554$		0			0
$555$	−1120.48	+	1120.48i	−0.0856965	+	0.0856965i
$556$		0			0
$557$	12780.7	−	12780.7i	0.972234	−	0.972234i	−0.0273908	−	0.999625i	$-0.508720\pi$
$557$	12780.7	−	12780.7i	0.972234	−	0.972234i	0.999625	+	0.0273908i	$0.00871985\pi$
$558$		0			0
$559$	12737.8			0.963775
$560$		0			0
$561$	−22724.7			−1.71023
$562$		0			0
$563$	−1558.47	+	1558.47i	−0.116664	+	0.116664i	−0.763028	−	0.646365i	$-0.776288\pi$
$563$	−1558.47	+	1558.47i	−0.116664	+	0.116664i	0.646365	+	0.763028i	$0.276288\pi$
$564$		0			0
$565$	−4142.23	+	4142.23i	−0.308433	+	0.308433i
$566$		0			0
$567$	15939.0	−	4799.40i	1.18056	−	0.355478i
$568$		0			0
$569$			12183.5i			0.897645i	0.893621	+	0.448822i	$0.148156\pi$
$569$			12183.5i			0.897645i	−0.893621	+	0.448822i	$0.851844\pi$
$570$		0			0
$571$	7602.31	−	7602.31i	0.557175	−	0.557175i	−0.371327	−	0.928502i	$-0.621097\pi$
$571$	7602.31	−	7602.31i	0.557175	−	0.557175i	0.928502	+	0.371327i	$0.121097\pi$
$572$		0			0
$573$	3411.38	+	3411.38i	0.248713	+	0.248713i
$574$		0			0
$575$		−	7103.73i		−	0.515210i
$576$		0			0
$577$			1467.50i			0.105880i	0.998598	+	0.0529399i	$0.0168592\pi$
$577$			1467.50i			0.105880i	−0.998598	+	0.0529399i	$0.983141\pi$
$578$		0			0
$579$	−12995.0	+	12995.0i	−0.932737	+	0.932737i
$580$		0			0
$581$	11665.7	−	21717.9i	0.833005	−	1.55079i
$582$		0			0
$583$	−22107.4			−1.57049
$584$		0			0
$585$		−	6039.17i		−	0.426819i
$586$		0			0
$587$	−1473.48	−	1473.48i	−0.103607	−	0.103607i	0.653403	−	0.757010i	$-0.273341\pi$
$587$	−1473.48	−	1473.48i	−0.103607	−	0.103607i	−0.757010	+	0.653403i	$0.773341\pi$
$588$		0			0
$589$	3886.47	−	3886.47i	0.271883	−	0.271883i
$590$		0			0
$591$	−16281.2			−1.13320
$592$		0			0
$593$			24576.6i			1.70192i	0.525230	+	0.850960i	$0.323979\pi$
$593$			24576.6i			1.70192i	−0.525230	+	0.850960i	$0.676021\pi$
$594$		0			0
$595$	−17974.3	−	9654.89i	−1.23845	−	0.665230i
$596$		0			0
$597$	−4284.42	−	4284.42i	−0.293718	−	0.293718i
$598$		0			0
$599$	21649.6			1.47676			0.738379	−	0.674386i	$-0.235592\pi$
$599$	21649.6			1.47676			0.738379	+	0.674386i	$0.235592\pi$
$600$		0			0
$601$	−12330.4			−0.836884			−0.418442	−	0.908244i	$-0.637424\pi$
$601$	−12330.4			−0.836884			−0.418442	+	0.908244i	$0.637424\pi$
$602$		0			0
$603$	2926.87	−	2926.87i	0.197664	−	0.197664i
$604$		0			0
$605$	5200.02	+	5200.02i	0.349439	+	0.349439i
$606$		0			0
$607$	−1201.49			−0.0803412			−0.0401706	−	0.999193i	$-0.512790\pi$
$607$	−1201.49			−0.0803412			−0.0401706	+	0.999193i	$0.512790\pi$
$608$		0			0
$609$	−968.944	−	3217.90i	−0.0644722	−	0.214115i
$610$		0			0
$611$	1017.16	+	1017.16i	0.0673481	+	0.0673481i
$612$		0			0
$613$	−381.742	−	381.742i	−0.0251524	−	0.0251524i	0.694419	−	0.719571i	$-0.255662\pi$
$613$	−381.742	−	381.742i	−0.0251524	−	0.0251524i	−0.719571	+	0.694419i	$0.755662\pi$
$614$		0			0
$615$		−	22953.5i		−	1.50500i
$616$		0			0
$617$			1454.88i			0.0949293i	0.998873	+	0.0474646i	$0.0151141\pi$
$617$			1454.88i			0.0949293i	−0.998873	+	0.0474646i	$0.984886\pi$
$618$		0			0
$619$	10557.7	+	10557.7i	0.685544	+	0.685544i	0.961244	−	0.275700i	$-0.0889097\pi$
$619$	10557.7	+	10557.7i	0.685544	+	0.685544i	−0.275700	+	0.961244i	$0.588910\pi$
$620$		0			0
$621$	4918.55	+	4918.55i	0.317833	+	0.317833i
$622$		0			0
$623$	16181.4	−	4872.39i	1.04060	−	0.313336i
$624$		0			0
$625$	19505.6			1.24836
$626$		0			0
$627$	−21606.5	−	21606.5i	−1.37621	−	1.37621i
$628$		0			0
$629$	−966.057	+	966.057i	−0.0612388	+	0.0612388i
$630$		0			0
$631$	−23126.7			−1.45905			−0.729524	−	0.683955i	$-0.760259\pi$
$631$	−23126.7			−1.45905			−0.729524	+	0.683955i	$0.760259\pi$
$632$		0			0
$633$	9318.94			0.585142
$634$		0			0
$635$	22093.4	+	22093.4i	1.38071	+	1.38071i
$636$		0			0
$637$	−8590.18	−	1745.32i	−0.534310	−	0.108559i
$638$		0			0
$639$		−	18157.3i		−	1.12409i
$640$		0			0
$641$	−20722.5			−1.27689			−0.638446	−	0.769667i	$-0.720422\pi$
$641$	−20722.5			−1.27689			−0.638446	+	0.769667i	$0.720422\pi$
$642$		0			0
$643$	−6238.25	+	6238.25i	−0.382601	+	0.382601i	−0.872038	−	0.489437i	$-0.837202\pi$
$643$	−6238.25	+	6238.25i	−0.382601	+	0.382601i	0.489437	+	0.872038i	$0.337202\pi$
$644$		0			0
$645$	−32452.5	−	32452.5i	−1.98111	−	1.98111i
$646$		0			0
$647$			16207.7i			0.984839i	0.870358	+	0.492420i	$0.163887\pi$
$647$			16207.7i			0.984839i	−0.870358	+	0.492420i	$0.836113\pi$
$648$		0			0
$649$	−18431.9			−1.11482
$650$		0			0
$651$	−2994.14	+	5574.14i	−0.180261	+	0.335588i
$652$		0			0
$653$	1458.15	−	1458.15i	0.0873842	−	0.0873842i	−0.662064	−	0.749448i	$-0.730319\pi$
$653$	1458.15	−	1458.15i	0.0873842	−	0.0873842i	0.749448	+	0.662064i	$0.230319\pi$
$654$		0			0
$655$		−	31490.6i		−	1.87853i
$656$		0			0
$657$			19088.4i			1.13350i
$658$		0			0
$659$	15058.7	+	15058.7i	0.890145	+	0.890145i	0.994536	−	0.104392i	$-0.0332896\pi$
$659$	15058.7	+	15058.7i	0.890145	+	0.890145i	−0.104392	+	0.994536i	$0.533290\pi$
$660$		0			0
$661$	8960.17	−	8960.17i	0.527247	−	0.527247i	−0.392504	−	0.919750i	$-0.628391\pi$
$661$	8960.17	−	8960.17i	0.527247	−	0.527247i	0.919750	+	0.392504i	$0.128391\pi$
$662$		0			0
$663$		−	13462.5i		−	0.788595i
$664$		0			0
$665$	−7910.08	−	26269.7i	−0.461263	−	1.53187i
$666$		0			0
$667$	2033.00	−	2033.00i	0.118018	−	0.118018i
$668$		0			0
$669$	−12806.9	+	12806.9i	−0.740125	+	0.740125i
$670$		0			0
$671$	14632.1			0.841829
$672$		0			0
$673$	1664.78			0.0953530			0.0476765	−	0.998863i	$-0.484818\pi$
$673$	1664.78			0.0953530			0.0476765	+	0.998863i	$0.484818\pi$
$674$		0			0
$675$	3160.97	−	3160.97i	0.180246	−	0.180246i
$676$		0			0
$677$	9944.97	−	9944.97i	0.564574	−	0.564574i	−0.366030	−	0.930603i	$-0.619283\pi$
$677$	9944.97	−	9944.97i	0.564574	−	0.564574i	0.930603	+	0.366030i	$0.119283\pi$
$678$		0			0
$679$	−6188.24	−	20551.4i	−0.349754	−	1.16155i
$680$		0			0
$681$			27753.4i			1.56169i
$682$		0			0
$683$	−2814.53	+	2814.53i	−0.157679	+	0.157679i	−0.781538	−	0.623858i	$-0.785564\pi$
$683$	−2814.53	+	2814.53i	−0.157679	+	0.157679i	0.623858	+	0.781538i	$0.285564\pi$
$684$		0			0
$685$	−4217.57	−	4217.57i	−0.235248	−	0.235248i
$686$		0			0
$687$		−	6343.90i		−	0.352307i
$688$		0			0
$689$		−	13096.8i		−	0.724160i
$690$		0			0
$691$	11213.0	−	11213.0i	0.617311	−	0.617311i	−0.327530	−	0.944841i	$-0.606216\pi$
$691$	11213.0	−	11213.0i	0.617311	−	0.617311i	0.944841	+	0.327530i	$0.106216\pi$
$692$		0			0
$693$	11985.6	+	6438.06i	0.656993	+	0.352903i
$694$		0			0
$695$	29048.7			1.58544
$696$		0			0
$697$		−	19790.1i		−	1.07547i
$698$		0			0
$699$	−8109.14	−	8109.14i	−0.438792	−	0.438792i
$700$		0			0
$701$	10576.9	−	10576.9i	0.569877	−	0.569877i	−0.362217	−	0.932094i	$-0.617980\pi$
$701$	10576.9	−	10576.9i	0.569877	−	0.569877i	0.932094	+	0.362217i	$0.117980\pi$
$702$		0			0
$703$	−1837.04			−0.0985568
$704$		0			0
$705$		−	5182.88i		−	0.276878i
$706$		0			0
$707$	−5758.93	+	10721.3i	−0.306346	+	0.570319i
$708$		0			0
$709$	−4578.76	−	4578.76i	−0.242537	−	0.242537i	0.575362	−	0.817899i	$-0.304861\pi$
$709$	−4578.76	−	4578.76i	−0.242537	−	0.242537i	−0.817899	+	0.575362i	$0.804861\pi$
$710$		0			0
$711$	13123.8			0.692237
$712$		0			0
$713$	−5413.26			−0.284331
$714$		0			0
$715$	−10817.7	+	10817.7i	−0.565817	+	0.565817i
$716$		0			0
$717$	−13236.7	−	13236.7i	−0.689445	−	0.689445i
$718$		0			0
$719$	15868.4			0.823078			0.411539	−	0.911392i	$-0.364991\pi$
$719$	15868.4			0.823078			0.411539	+	0.911392i	$0.364991\pi$
$720$		0			0
$721$	9780.97	+	32483.0i	0.505218	+	1.67785i
$722$		0			0
$723$	−1937.94	−	1937.94i	−0.0996859	−	0.0996859i
$724$		0			0
$725$	−1306.54	−	1306.54i	−0.0669291	−	0.0669291i
$726$		0			0
$727$		−	9436.13i		−	0.481385i	−0.970601	−	0.240692i	$-0.922626\pi$
$727$		−	9436.13i		−	0.481385i	0.970601	−	0.240692i	$-0.0773745\pi$
$728$		0			0
$729$		−	3452.58i		−	0.175409i
$730$		0			0
$731$	−27980.0	−	27980.0i	−1.41570	−	1.41570i
$732$		0			0
$733$	−13472.0	−	13472.0i	−0.678855	−	0.678855i	0.280886	−	0.959741i	$-0.409372\pi$
$733$	−13472.0	−	13472.0i	−0.678855	−	0.678855i	−0.959741	+	0.280886i	$0.909372\pi$
$734$		0			0
$735$	17438.9	+	26332.1i	0.875160	+	1.32146i
$736$		0			0
$737$	−10485.5			−0.524070
$738$		0			0
$739$	−8172.74	−	8172.74i	−0.406819	−	0.406819i	0.473809	−	0.880628i	$-0.342879\pi$
$739$	−8172.74	−	8172.74i	−0.406819	−	0.406819i	−0.880628	+	0.473809i	$0.842879\pi$
$740$		0			0
$741$	12800.0	−	12800.0i	0.634576	−	0.634576i
$742$		0			0
$743$	−5760.56			−0.284434			−0.142217	−	0.989836i	$-0.545423\pi$
$743$	−5760.56			−0.284434			−0.142217	+	0.989836i	$0.545423\pi$
$744$		0			0
$745$	30353.2			1.49269
$746$		0			0
$747$	−16028.6	−	16028.6i	−0.785082	−	0.785082i
$748$		0			0
$749$	−3718.01	+	6921.76i	−0.181380	+	0.337671i
$750$		0			0
$751$		−	14758.8i		−	0.717119i	−0.933507	−	0.358560i	$-0.883268\pi$
$751$		−	14758.8i		−	0.717119i	0.933507	−	0.358560i	$-0.116732\pi$
$752$		0			0
$753$	48124.6			2.32903
$754$		0			0
$755$	−4570.01	+	4570.01i	−0.220291	+	0.220291i
$756$		0			0
$757$	−4101.19	−	4101.19i	−0.196909	−	0.196909i	0.601765	−	0.798674i	$-0.294465\pi$
$757$	−4101.19	−	4101.19i	−0.196909	−	0.196909i	−0.798674	+	0.601765i	$0.794465\pi$
$758$		0			0
$759$			30094.6i			1.43921i
$760$		0			0
$761$	21588.7			1.02837			0.514184	−	0.857680i	$-0.328095\pi$
$761$	21588.7			1.02837			0.514184	+	0.857680i	$0.328095\pi$
$762$		0			0
$763$	−4995.57	+	9300.15i	−0.237027	+	0.441269i
$764$		0			0
$765$	−13265.8	+	13265.8i	−0.626960	+	0.626960i
$766$		0			0
$767$		−	10919.3i		−	0.514047i
$768$		0			0
$769$		−	16311.0i		−	0.764878i	−0.923981	−	0.382439i	$-0.875084\pi$
$769$		−	16311.0i		−	0.764878i	0.923981	−	0.382439i	$-0.124916\pi$
$770$		0			0
$771$	17071.7	+	17071.7i	0.797436	+	0.797436i
$772$		0			0
$773$	5816.48	−	5816.48i	0.270639	−	0.270639i	−0.558718	−	0.829358i	$-0.688707\pi$
$773$	5816.48	−	5816.48i	0.270639	−	0.270639i	0.829358	+	0.558718i	$0.188707\pi$
$774$		0			0
$775$			3478.90i			0.161246i
$776$		0			0
$777$	2025.01	−	609.751i	0.0934966	−	0.0281528i
$778$		0			0
$779$	18816.3	−	18816.3i	0.865424	−	0.865424i
$780$		0			0
$781$	−32524.4	+	32524.4i	−1.49016	+	1.49016i
$782$		0			0
$783$	1809.27			0.0825771
$784$		0			0
$785$	8106.88			0.368595
$786$		0			0
$787$	−6614.00	+	6614.00i	−0.299572	+	0.299572i	−0.840846	−	0.541274i	$-0.817942\pi$
$787$	−6614.00	+	6614.00i	−0.299572	+	0.299572i	0.541274	+	0.840846i	$0.317942\pi$
$788$		0			0
$789$	1622.14	−	1622.14i	0.0731937	−	0.0731937i
$790$		0			0
$791$	7486.16	−	2254.16i	0.336507	−	0.101326i
$792$		0			0
$793$			8668.29i			0.388172i
$794$		0			0
$795$	−33367.0	+	33367.0i	−1.48856	+	1.48856i
$796$		0			0
$797$	−15234.0	−	15234.0i	−0.677058	−	0.677058i	0.282275	−	0.959333i	$-0.408911\pi$
$797$	−15234.0	−	15234.0i	−0.677058	−	0.677058i	−0.959333	+	0.282275i	$0.908911\pi$
$798$		0			0
$799$		−	4468.60i		−	0.197857i
$800$		0			0
$801$		−	15538.5i		−	0.685426i
$802$		0			0
$803$	34192.1	−	34192.1i	1.50263	−	1.50263i
$804$		0			0
$805$	−12786.1	+	23803.6i	−0.559813	+	1.04219i
$806$		0			0
$807$	34577.7			1.50829
$808$		0			0
$809$			18106.2i			0.786872i	0.919352	+	0.393436i	$0.128714\pi$
$809$			18106.2i			0.786872i	−0.919352	+	0.393436i	$0.871286\pi$
$810$		0			0
$811$	−4279.91	−	4279.91i	−0.185312	−	0.185312i	0.608354	−	0.793666i	$-0.291830\pi$
$811$	−4279.91	−	4279.91i	−0.185312	−	0.185312i	−0.793666	+	0.608354i	$0.791830\pi$
$812$		0			0
$813$	36665.0	−	36665.0i	1.58167	−	1.58167i
$814$		0			0
$815$	−27203.3			−1.16919
$816$		0			0
$817$		−	53206.5i		−	2.27841i
$818$		0			0
$819$	−3814.00	+	7100.46i	−0.162725	+	0.302943i
$820$		0			0
$821$	−14453.4	−	14453.4i	−0.614406	−	0.614406i	0.329685	−	0.944091i	$-0.393057\pi$
$821$	−14453.4	−	14453.4i	−0.614406	−	0.614406i	−0.944091	+	0.329685i	$0.893057\pi$
$822$		0			0
$823$	−32668.8			−1.38367			−0.691837	−	0.722053i	$-0.743198\pi$
$823$	−32668.8			−1.38367			−0.691837	+	0.722053i	$0.743198\pi$
$824$		0			0
$825$	19340.7			0.816189
$826$		0			0
$827$	−5700.90	+	5700.90i	−0.239709	+	0.239709i	−0.816730	−	0.577020i	$-0.804215\pi$
$827$	−5700.90	+	5700.90i	−0.239709	+	0.239709i	0.577020	+	0.816730i	$0.304215\pi$
$828$		0			0
$829$	16310.4	+	16310.4i	0.683335	+	0.683335i	0.960750	−	0.277415i	$-0.0894777\pi$
$829$	16310.4	+	16310.4i	0.683335	+	0.683335i	−0.277415	+	0.960750i	$0.589478\pi$
$830$		0			0
$831$	−25910.4			−1.08162
$832$		0			0
$833$	15035.5	+	22703.2i	0.625391	+	0.944319i
$834$		0			0
$835$	9889.24	+	9889.24i	0.409858	+	0.409858i
$836$		0			0
$837$	−2408.76	−	2408.76i	−0.0994729	−	0.0994729i
$838$		0			0
$839$			5744.91i			0.236396i	0.992990	+	0.118198i	$0.0377118\pi$
$839$			5744.91i			0.236396i	−0.992990	+	0.118198i	$0.962288\pi$
$840$		0			0
$841$			23641.2i			0.969337i
$842$		0			0
$843$	28298.0	+	28298.0i	1.15615	+	1.15615i
$844$		0			0
$845$	15149.3	+	15149.3i	0.616748	+	0.616748i
$846$		0			0
$847$	−2829.80	−	9397.88i	−0.114797	−	0.381245i
$848$		0			0
$849$	26972.7			1.09034
$850$		0			0
$851$	1279.36	+	1279.36i	0.0515345	+	0.0515345i
$852$		0			0
$853$	−2808.64	+	2808.64i	−0.112738	+	0.112738i	−0.761226	−	0.648487i	$-0.775402\pi$
$853$	−2808.64	+	2808.64i	−0.112738	+	0.112738i	0.648487	+	0.761226i	$0.275402\pi$
$854$		0			0
$855$	−25226.0			−1.00902
$856$		0			0
$857$	−26003.3			−1.03647			−0.518236	−	0.855237i	$-0.673411\pi$
$857$	−26003.3			−1.03647			−0.518236	+	0.855237i	$0.673411\pi$
$858$		0			0
$859$	−15888.8	−	15888.8i	−0.631105	−	0.631105i	0.317240	−	0.948345i	$-0.397244\pi$
$859$	−15888.8	−	15888.8i	−0.631105	−	0.631105i	−0.948345	+	0.317240i	$0.897244\pi$
$860$		0			0
$861$	−14496.1	+	26987.2i	−0.573782	+	1.06820i
$862$		0			0
$863$		−	24922.9i		−	0.983066i	−0.870859	−	0.491533i	$-0.836437\pi$
$863$		−	24922.9i		−	0.983066i	0.870859	−	0.491533i	$-0.163563\pi$
$864$		0			0
$865$	27669.3			1.08761
$866$		0			0
$867$	−6520.21	+	6520.21i	−0.255407	+	0.255407i
$868$		0			0
$869$	−23508.1	−	23508.1i	−0.917671	−	0.917671i
$870$		0			0
$871$		−	6211.78i		−	0.241651i
$872$		0			0
$873$	−19734.9			−0.765092
$874$		0			0
$875$	−13003.3	−	6984.68i	−0.502389	−	0.269858i
$876$		0			0
$877$	31437.3	−	31437.3i	1.21045	−	1.21045i	0.239569	−	0.970879i	$-0.422994\pi$
$877$	31437.3	−	31437.3i	1.21045	−	1.21045i	0.970879	−	0.239569i	$-0.0770060\pi$
$878$		0			0
$879$			29745.4i			1.14140i
$880$		0			0
$881$			26911.4i			1.02913i	0.857450	+	0.514567i	$0.172047\pi$
$881$			26911.4i			1.02913i	−0.857450	+	0.514567i	$0.827953\pi$
$882$		0			0
$883$	21407.8	+	21407.8i	0.815889	+	0.815889i	0.985509	−	0.169620i	$-0.0542542\pi$
$883$	21407.8	+	21407.8i	0.815889	+	0.815889i	−0.169620	+	0.985509i	$0.554254\pi$
$884$		0			0
$885$	−27819.6	+	27819.6i	−1.05666	+	1.05666i
$886$		0			0
$887$			33419.9i			1.26509i	0.774525	+	0.632543i	$0.217989\pi$
$887$			33419.9i			1.26509i	−0.774525	+	0.632543i	$0.782011\pi$
$888$		0			0
$889$	−12023.0	−	39928.9i	−0.453587	−	1.50638i
$890$		0			0
$891$	−27416.5	+	27416.5i	−1.03085	+	1.03085i
$892$		0			0
$893$	4248.72	−	4248.72i	0.159214	−	0.159214i
$894$		0			0
$895$	37223.2			1.39021
$896$		0			0
$897$	−17828.5			−0.663629
$898$		0			0
$899$	−995.621	+	995.621i	−0.0369364	+	0.0369364i
$900$		0			0
$901$	−28768.6	+	28768.6i	−1.06373	+	1.06373i
$902$		0			0
$903$	17660.3	+	58650.6i	0.650828	+	2.16143i
$904$		0			0
$905$			38477.5i			1.41330i
$906$		0			0
$907$	−14086.4	+	14086.4i	−0.515689	+	0.515689i	−0.916264	−	0.400575i	$-0.868810\pi$
$907$	−14086.4	+	14086.4i	−0.515689	+	0.515689i	0.400575	+	0.916264i	$0.368810\pi$
$908$		0			0
$909$	7912.73	+	7912.73i	0.288722	+	0.288722i
$910$		0			0
$911$			19973.6i			0.726404i	0.931711	+	0.363202i	$0.118316\pi$
$911$			19973.6i			0.726404i	−0.931711	+	0.363202i	$0.881684\pi$
$912$		0			0
$913$			57422.7i			2.08150i
$914$		0			0
$915$	22084.5	−	22084.5i	0.797914	−	0.797914i
$916$		0			0
$917$	−19887.7	+	37024.5i	−0.716193	+	1.33332i
$918$		0			0
$919$	−40573.4			−1.45636			−0.728179	−	0.685387i	$-0.759633\pi$
$919$	−40573.4			−1.45636			−0.728179	+	0.685387i	$0.759633\pi$
$920$		0			0
$921$		−	7552.74i		−	0.270218i
$922$		0			0
$923$	−19267.9	−	19267.9i	−0.687119	−	0.687119i
$924$		0			0
$925$	822.197	−	822.197i	0.0292256	−	0.0292256i
$926$		0			0
$927$	31192.5			1.10517
$928$		0			0
$929$		−	27431.3i		−	0.968775i	−0.874854	−	0.484387i	$-0.839043\pi$
$929$		−	27431.3i		−	0.968775i	0.874854	−	0.484387i	$-0.160957\pi$
$930$		0			0
$931$	−7290.33	+	35881.7i	−0.256639	+	1.26313i
$932$		0			0
$933$	−14757.4	−	14757.4i	−0.517829	−	0.517829i
$934$		0			0
$935$	47524.7			1.66227
$936$		0			0
$937$	34087.6			1.18847			0.594234	−	0.804293i	$-0.297455\pi$
$937$	34087.6			1.18847			0.594234	+	0.804293i	$0.297455\pi$
$938$		0			0
$939$	22433.2	−	22433.2i	0.779637	−	0.779637i
$940$		0			0
$941$	18574.1	+	18574.1i	0.643462	+	0.643462i	0.951405	−	0.307943i	$-0.0996407\pi$
$941$	18574.1	+	18574.1i	0.643462	+	0.643462i	−0.307943	+	0.951405i	$0.599641\pi$
$942$		0			0
$943$	−26208.3			−0.905046
$944$		0			0
$945$	−16281.4	+	4902.50i	−0.560460	+	0.168760i
$946$		0			0
$947$	15534.3	+	15534.3i	0.533050	+	0.533050i	0.921479	−	0.388429i	$-0.126982\pi$
$947$	15534.3	+	15534.3i	0.533050	+	0.533050i	−0.388429	+	0.921479i	$0.626982\pi$
$948$		0			0
$949$	20255.9	+	20255.9i	0.692871	+	0.692871i
$950$		0			0
$951$		−	6215.41i		−	0.211933i
$952$		0			0
$953$			42060.4i			1.42967i	0.699296	+	0.714833i	$0.253497\pi$
$953$			42060.4i			1.42967i	−0.699296	+	0.714833i	$0.746503\pi$
$954$		0			0
$955$	−7134.28	−	7134.28i	−0.241738	−	0.241738i
$956$		0			0
$957$	5535.08	+	5535.08i	0.186963	+	0.186963i
$958$		0			0
$959$	2295.16	+	7622.31i	0.0772831	+	0.256660i
$960$		0			0
$961$	−27140.0			−0.911012
$962$		0			0
$963$	5108.53	+	5108.53i	0.170945	+	0.170945i
$964$		0			0
$965$	27176.7	−	27176.7i	0.906581	−	0.906581i
$966$		0			0
$967$	−2353.43			−0.0782638			−0.0391319	−	0.999234i	$-0.512459\pi$
$967$	−2353.43			−0.0782638			−0.0391319	+	0.999234i	$0.512459\pi$
$968$		0			0
$969$	−56233.5			−1.86427
$970$		0			0
$971$	9681.29	+	9681.29i	0.319966	+	0.319966i	0.848754	−	0.528788i	$-0.177353\pi$
$971$	9681.29	+	9681.29i	0.319966	+	0.319966i	−0.528788	+	0.848754i	$0.677353\pi$
$972$		0			0
$973$	−34153.5	−	18345.5i	−1.12529	−	0.604451i
$974$		0			0
$975$			11457.7i			0.376349i
$976$		0			0
$977$	7219.06			0.236395			0.118198	−	0.992990i	$-0.462288\pi$
$977$	7219.06			0.236395			0.118198	+	0.992990i	$0.462288\pi$
$978$		0			0
$979$	−27833.4	+	27833.4i	−0.908642	+	0.908642i
$980$		0			0
$981$	6863.87	+	6863.87i	0.223391	+	0.223391i
$982$		0			0
$983$		−	6107.36i		−	0.198163i	−0.995079	−	0.0990817i	$-0.968409\pi$
$983$		−	6107.36i		−	0.198163i	0.995079	−	0.0990817i	$-0.0315905\pi$
$984$		0			0
$985$	34049.3			1.10142
$986$		0			0
$987$	−3273.22	+	6093.69i	−0.105560	+	0.196519i
$988$		0			0
$989$	−37054.2	+	37054.2i	−1.19136	+	1.19136i
$990$		0			0
$991$			28231.6i			0.904951i	0.891777	+	0.452476i	$0.149459\pi$
$991$			28231.6i			0.904951i	−0.891777	+	0.452476i	$0.850541\pi$
$992$		0			0
$993$		−	18265.2i		−	0.583715i
$994$		0			0
$995$	8960.09	+	8960.09i	0.285481	+	0.285481i
$996$		0			0
$997$	13633.8	−	13633.8i	0.433085	−	0.433085i	−0.456592	−	0.889676i	$-0.650930\pi$
$997$	13633.8	−	13633.8i	0.433085	−	0.433085i	0.889676	+	0.456592i	$0.150930\pi$
$998$		0			0
$999$			1138.56i			0.0360586i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.j.b.335.35		88
4.3	odd	2		112.4.j.b.27.13	✓	88
7.6	odd	2	inner	448.4.j.b.335.10		88
16.3	odd	4	inner	448.4.j.b.111.10		88
16.13	even	4		112.4.j.b.83.14	yes	88
28.27	even	2		112.4.j.b.27.14	yes	88
112.13	odd	4		112.4.j.b.83.13	yes	88
112.83	even	4	inner	448.4.j.b.111.35		88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.j.b.27.13	✓	88	4.3	odd	2
112.4.j.b.27.14	yes	88	28.27	even	2
112.4.j.b.83.13	yes	88	112.13	odd	4
112.4.j.b.83.14	yes	88	16.13	even	4
448.4.j.b.111.10		88	16.3	odd	4	inner
448.4.j.b.111.35		88	112.83	even	4	inner
448.4.j.b.335.10		88	7.6	odd	2	inner
448.4.j.b.335.35		88	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.69197 - 4.69197i) q^{3} +(-9.81242 + 9.81242i) q^{5} +(17.7338 - 5.33982i) q^{7} -17.0292i q^{9} +(-30.5036 + 30.5036i) q^{11} +(-18.0708 - 18.0708i) q^{13} +92.0792i q^{15} +79.3893i q^{17} +(-75.4829 + 75.4829i) q^{19} +(58.1520 - 108.261i) q^{21} +105.136 q^{23} -67.5671i q^{25} +(46.7827 + 46.7827i) q^{27} +(19.3369 - 19.3369i) q^{29} -51.4882 q^{31} +286.244i q^{33} +(-121.615 + 226.408i) q^{35} +(12.1686 + 12.1686i) q^{37} -169.575 q^{39} -249.280 q^{41} +(-352.441 + 352.441i) q^{43} +(167.098 + 167.098i) q^{45} -56.2873 q^{47} +(285.973 - 189.390i) q^{49} +(372.492 + 372.492i) q^{51} +(362.373 + 362.373i) q^{53} -598.629i q^{55} +708.327i q^{57} +(302.126 + 302.126i) q^{59} +(-239.843 - 239.843i) q^{61} +(-90.9328 - 301.992i) q^{63} +354.636 q^{65} +(171.874 + 171.874i) q^{67} +(493.295 - 493.295i) q^{69} +1066.25 q^{71} -1120.92 q^{73} +(-317.023 - 317.023i) q^{75} +(-378.060 + 703.828i) q^{77} +770.664i q^{79} +898.795 q^{81} +(941.243 - 941.243i) q^{83} +(-779.000 - 779.000i) q^{85} -181.456i q^{87} +912.463 q^{89} +(-416.958 - 223.968i) q^{91} +(-241.581 + 241.581i) q^{93} -1481.34i q^{95} -1158.89i q^{97} +(519.452 + 519.452i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77}+ \cdots - 7592 q^{99}+O(q^{100}) \) 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Introduction

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Modular forms

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